18 CANONICAL DECOMPOSITIONS OF A MEASURE 205 mapping U is said to be of type (p, q) if U(E) c LJ(Y, v) and if U is continuous in the topology induced on E by that of L£(X, /x), U being considered as a mapping of E into L£(Y, v). Let \\U\\ptq denote the norm of this linear mapping. If p =£ +00, then U may be extended by continuity to a linear mapping, with norm \\U\\pt q , of Lg(X, JJL) into LJ(Y, v). The same is true if p = 4- oo, provided that p, is bounded. Show that if U is of type (p0 , q0) and also of type Oi, #0, then £/ is also of type (p, q) where 1 __ 1 -t t 1 _ 1 -t t P Po Pi' q Qo <?i for all t e [0, 1] (with the convention that l/oo =0, 1/0= +00); moreover, we have \\u\\,. 9z ii oiij-v iiWpi.«i (M. Riesz-Thorin interpolation theorem). (Use Problem 1, by majorizing the integral \ \(U '/)#! dv for /e E and g e L£(Y", v), where - + — = 1 ; for this purpose, put J ~l T /= I/IK, 0= 101" with H = \v\ = 1, and /c= |/|;w, ^c= \g\^v for all complex numbers £. Then choose appropriately two affine-linear functions and apply the three-line theorem (Problem 6) to the holomorphic function 18. CANONICAL DECOMPOSITIONS OF A MEASURE Two (complex) measures ju, v on X are said to be disjoint if inf(|^|, |v|) = 0. A. measure JJL is said to be concentrated on a set M, or carried by M, if X — M is \^-negligible, or equivalent^ if |ju| = <pM • |ju| (13.15.3); every measure with base |ju| is then also concentrated on M (13.15.5). (13.18.1) For two measures ju, v to be disjoint it is necessary and sufficient that there should exist two disjoint subsets M, N ofX such that ju is concentrated on M and v is concentrated on N; and the sets M, N can be chosen to be universally measurable. We may restrict ourselves to the case where ^ and v are positive, and then we can write JJL — g * p and v = h • p, where p is a positve measure and g, h are locally p-integrable functions ;>0 ((13.15.2) and (13.15.3)). We have then inf(/i, v) = inf(#, h) - p, and hence inf^u, v) = 0 if and only if inf(#, h) is p-negligible (13.15.3). If M0 and N0 are the sets of points xeX such that g(x) ^ 0 and h(x) ^ 0, respectively, then inf(#, h) is p-negligible if and onlygmen-Lindelof principle as in Section 9.5, Problem 17, by taking g(s) = e*2.)